| Abstract: |
| We state and analyze nonlocal problems with classically-defined, local boundary conditions. The model takes its horizon parameter to be spatially dependent, vanishing near the boundary of the domain. We establish a Green`s identity for the nonlocal operator that recovers the classical boundary integral, which permits the use of variational techniques. Using this, we show the existence of weak solutions, as well as their variational convergence to classical counterparts as the horizon uniformly converges to zero. In certain circumstances, global regularity of solutions can be established, resulting in improved modes and rates of variational convergence. We also show that Galerkin discretization schemes for the nonlocal problems converge unconditionally with respect to the nonlocal parameter, i.e. that the schemes are asymptotically compatible. |
|